By D. Chatterjee
In regards to the publication: This booklet offers exposition of the topic either in its common and algebraic facets. It offers with the notions of topological areas, compactness, connectedness, completeness together with metrizability and compactification, algebraic elements of topological areas via homotopy teams and homology teams. It starts with the fundamental notions of topological areas yet quickly going past them reaches the Read more...
summary: in regards to the ebook: This publication presents exposition of the topic either in its common and algebraic facets. It offers with the notions of topological areas, compactness, connectedness, completeness together with metrizability and compactification, algebraic elements of topological areas via homotopy teams and homology teams. It starts off with the elemental notions of topological areas yet quickly going past them reaches the area of algebra in the course of the notions of homotopy, homology and cohomology. How those methods paintings in concord is the subject material of this booklet. The ebook eventually arrives on the
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Extra resources for Topology : general & algebraic
Example text
For example the interval (0, 1] is neither open nor closed in R when equipped with the usual topology. Every subset of a discrete metric space is both open and closed. 40 Topology The following theorem is an analogue of the corresponding results on open sets. 4: The following are true in a metric space (M, d): (i) φ and M are closed (ii) The intersection of an arbirary family of closed sets is closed (iii) The union of a finite family of closed sets is closed. Proof: (i) Trivial. (ii) Let {Fα }α ∈Λ be an arbitrary family of closed sets.
Then x ∈ K ′ iff there exists a sequence xn in K such that lim xn = x . Proof: Let x ∈ K ′. Then for every δ > 0, Sδ ( x ) ∩ K ≠ φ . So if we choose δ = 1/n, we get a sequence xn in Sδ ( x ) ∩ K such that d ( xn , x ) < 1/ n. This is our desired xn which lies in K and converges to x. Conversely, let xn be a sequence in K, converging to x. We shall show that x ∈ K ′. By definition, for every ε > 0, there exists a positive integer n0 such that d ( xn , x ) < ε for all n ≥ n0 . Thus for every ε > 0, Sε ( x ) ∩ K ≠ φ .
Analogously, ( R n , d ) is separable when d is the usual metric in R n . Note a discrete metric space is separable if and only if it is countable. Thus an uncountable set with its discrete topology is not separable. Metric Space 41 Definition: A point p is called an exterior point of a set K in a metric space (M, d) if it is an interior point of K c . The set of exterior points of a set K is called the exterior of K and is usually denoted by ext (K). , q is a limit point of both K and K c . The set of all boundary points of a set K is called the (topological) boundary of K and is usually denoted by bdry (K) or ∂ K.